A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods

Transcription

1 Advances n Appled Mathematcs and Mechancs Adv. Appl. Math. Mech., Vol., No., pp. -9 DOI:.8/aamm..m June A Comparson of the Performance of Lmters for Runge-Kutta Dscontnuous Galerkn Methods Hongqang Zhu, Yue Cheng, and Janan Qu, School of Natural Scence, Nanjng Unversty of Posts and Telecommuncatons, Nanjng, Jangsu, Chna Department of Mathematcs, Nanjng Unversty, Nanjng, Jangsu 9, Chna School of Mathematcal Scences, Xamen Unversty, Xamen, Fujan, Chna Badu, Inc. Badu Campus, No., Shangd th Street, Hadan Dstrct, Bejng 8, Chna Receved November ; Accepted (n revsed verson) 8 February Avalable onlne Aprl Abstract. Dscontnutes usually appear n solutons of nonlnear conservaton laws even though the ntal condton s smooth, whch leads to great dffculty n computng these solutons numercally. The Runge-Kutta dscontnuous Galerkn (RKDG) methods are effcent methods for solvng nonlnear conservaton laws, whch are hghorder accurate and hghly parallelzable, and can be easly used to handle complcated geometres and boundary condtons. An mportant component of RKDG methods for solvng nonlnear conservaton laws wth strong dscontnutes n the soluton s a nonlnear lmter, whch s appled to detect dscontnutes and control spurous oscllatons near such dscontnutes. Many such lmters have been used n the lterature on RKDG methods. A lmter contans two parts, frst to dentfy the troubled cells, namely, those cells whch mght need the lmtng procedure, then to replace the soluton polynomals n those troubled cells by reconstructed polynomals whch mantan the orgnal cell averages (conservaton). [SIAM J. Sc. Comput., (), pp. 99.] focused on dscussng the frst part of lmters. In ths paper, focused on the second part, we wll systematcally nvestgate and compare a few dfferent reconstructon strateges wth an objectve of obtanng the most effcent and relable reconstructon strategy. Ths work can help wth the choosng of rght lmters so one can resolve sharper dscontnutes, get better numercal solutons and save the computatonal cost. AMS subject classfcatons: M, M99, L Key words: Lmter, dscontnuous Galerkn method, hyperbolc conservaton laws. Correspondng author. URL: Emal: zhuhq@njupt.edu.cn (H. Zhu), chengyue@gmal.com (Y. Cheng), jqu@mu.edu.cn (J. Qu) c Global Scence Press

2 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 Introducton The Runge-Kutta dscontnuous Galerkn (RKDG) methods for solvng hyperbolc conservaton laws are hgh-order accurate and hghly parallelzable methods whch can easly handle complcated geometres and boundary condtons. These methods have made ther way nto the man stream of computatonal flud dynamcs and other areas of applcatons. The frst DG method was ntroduced n 9 by Reed and Hll [] for the neutron transport problem. A major development of ths method was carred out by Cockburn et al. n a seres of papers [ ], n whch a framework to solve nonlnear tme dependent hyperbolc conservaton laws was establshed. They adopted eplct, nonlnearly stable hgh order Runge-Kutta tme dscretzatons [8], DG space dscretzatons wth eact or appromate Remann solvers as nterface flues and TVB (total varaton bounded) nonlnear lmter [] to acheve nonoscllatory propertes, and the method was termed as RKDG method. We wll brefly revew ths method n Secton. Detaled descrpton of the method as well as ts mplementaton can be found n the revew paper [8]. Solutons of nonlnear hyperbolc conservaton laws usually have dscontnutes even though the ntal condtons are smooth, whch leads to great dffculty n computng these solutons numercally. An mportant component of RKDG methods for solvng conservaton laws wth strong shocks n the soluton s a nonlnear lmter, whch s appled to detect dscontnutes and control spurous oscllatons near such dscontnutes. Many such lmters have been used n the RKDG methods. Cockburn et al. developed the mnmod-type TVB lmter [ ], whch s a slope lmter usng a technque borrowed from the fnte volume methodology. Bswas et al. proposed a moment lmter [] whch s specfcally desgned for DG methods and works on the moments of the numercal soluton. Ths moment lmter was later mproved by Burbeau et al. [] and mproved further by Krvodonova []. There are also many lmters developed n the fnte volume and fnte dfference lterature, such as varous flu lmters [], monotoncty-preservng (MP) lmters [] and modfed MP lmters []. Although there are many lmters that we can use n the RKDG methods, none of them s reported to be obvously better than the others for varous problems. Numercal eperments n the lterature tell that dfferent lmters usually behave dfferently for the same problem and the same lmter may behave dfferently for dfferent problems. There s no gudelne for people to choose a rght lmter for a certan problem. So a systematc study of lmters s necessary. Qu and Shu [] adopted a new framework to devse a lmter for the RKDG methods. They dvded a lmter nto two separate parts. The frst part s a troubled-cell ndcator, whch s a dscontnuty detecton strategy whch detects the cells that are beleved to contan a dscontnuty and need the lmtng procedure. The second part s a soluton reconstructon method whch s appled only on the detected cells. The troubledcell ndcators can come from any lmters or shock detectng technques. Focused on the frst part of lmters, Qu and Shu [] presented an overvew of the troubled-cell ndcators and made a comparson of ther performance n conjuncton wth a hgh-order

3 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 WENO (weghted essentally nonoscllatory) soluton reconstructon method. In ths paper, focused on the second part, we wll systematcally nvestgate and compare a few dfferent reconstructon strateges wth an objectve of obtanng the most effcent and relable reconstructon strategy. Ths work can help wth the choosng of rght lmters so one can resolve sharper dscontnutes, get better numercal solutons and save the computatonal cost. The outlne of the remander of the paper s as follows. In Secton we brefly revew the RKDG methods n one dmenson. Secton revews the lmters used n ths paper. Numercal comparsons and computatonal results on a varety of test cases are presented n Secton wth conclusons followng n Secton. Revew of RKDG method We consder the one-dmensonal scalar conservaton law { ut f(u) =, u(,)=u (). (.) We dvde the computatonal doman [,L] nto N cells wth boundary ponts = < < < N = L. Denote the center of cell I =[, ] by, and the length of cell I by. The soluton as well as the test functon space s gven by V k h ={p:p I P k (I )}, where P k (I ) s the space of polynomals of degree at most k on cell I. We adopt the Legendre polynomals W ()=, W l ()= d l ( ) l l l! d l, l=,,k, (.) as the local bass functons. However, we emphasze that the procedure descrbed below does not depend on the specfc bass chosen for the polynomals and works also n multple dmensons. The Legendre polynomals are L -orthogonal, namely, W l ()W l ()d= l, f l= l,, otherwse. And now we can epress our appromate soluton u h as follows: u h (,t)= k l= u (l) (t)v () l () for I, (.) where v () l ()=W l (( )/ ),

4 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 and u (l) (t) (l =,,k) are the degrees of freedom. We wll omt the argument t and denote them as u (l) n ths paper. In order to determne the appromate soluton, we multply (.) by test functons v () l () (l=,,k), ntegrate over cell I, ntegrate by parts, and we are able to evolve the degrees of freedom u (l) : d l dt u(l) f(u h (,t)) dv () l () d f(u I d ˆ,u ) ( ) l ˆf(u,u l=,,k, (.) )=, where u ± / =uh ( ± /,t) are the left and rght lmts of the dscontnuous soluton uh at the cell nterface /, and ˆf(u,u ) s a consstent and monotone (nondecreasng n the frst argument and nonncreasng n the second argument) flu for the scalar case and an eact or appromate Remann solver for the system case. The semdscrete scheme (.) s an ODE system. One dscretzes t usng the total varaton dmnshng (TVD) Runge- Kutta tme dscretzaton ntroduced n [8], whch completes the defnton of RKDG method. In ths paper for k=, we use the second order Runge-Kutta tme steppng φ () = φ n tl(φ n ), φ n = φn φ() tl(φ() ), (.a) (.b) f the ODE system s denoted by φ t = L(φ). For k=, we use the followng thrd order verson φ () = φ n tl(φ n ), φ () = φn φ() tl(φ() ), φ n = φn φ() tl(φ() ). (.a) (.b) (.c) Descrpton of lmters In ths secton, we descrbe a few commonly used lmters whch are chosen for our numercal comparsons.. Mnmod-based TVB lmter (TVB lmter) []. Denote From (.) we can derve ũ = k l= u u (l) v () = u () ũ, u = u () ũ. (.) l ( ), ũ = k l= u (l) v () l ( ). (.)

5 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp They are modfed by the TVB-modfed mnmod functon ũ (mod) = m(ũ,u () u(),u () u () ), ũ (mod) = m(ũ,u () u(),u () u () ), (.a) (.b) where m s gven by { a, f a m(a,a,,a n )= Mh, m(a,a,,a n ), otherwse, (.) and the mnmod functon m s gven by { s mn j n a m(a,a,,a n )= j, f sgn(a )=sgn(a )= = sgn(a n )=s,, otherwse. (.) The parameter M> s a constant. Unfortunately, the TVB lmter constant M s dependent on the problem. There s no automatc swtchng whch works well for varous stuatons. As t was ponted out n [], the resoluton of soluton s dependent on the choce of the constant M; and sometmes, the case of k= may gve better resoluton to shocks or contact dscontnutes than the case of k= f an napproprate M s used. Tunng M requres much epermental research. But once we get the approprate M, the TVB lmter wll make the soluton much better. In ths paper, M s chosen based on our numercal trals and eperence and the correspondng lmter s denoted by TVB-.. Moment lmter of Bswas, Devne, and Flaherty (BDF lmter) []. The moments are modfed as follows: u (l),mod = ( l m (l )u (l),u (l ) u (l ),u (l ) ) u (l ), l k, (.) where m s agan the mnmod functon. Ths lmter s appled adaptvely. Frst, the hghest-order moment u (k) s lmted. Then the lmter s appled successvely to lowerorder moments when the net hgher-order moment has been changed by the lmtng.. A modfcaton of the moment lmter by Burbeau, Sagaut, and Bruneau (BSB lmter) []. We defne If u (l),m u (l),m = ( l m (l )u (l) = u (l), the moment u (l) s replaced wth u (l),mod,u (l ) u (l ),u (l ) = mamod ( u (l),m,u (l),ma ) u (l ), l k. (.) ), (.8)

6 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 where { s ma j n a mamod(a,a,,a n )= j, f sgn(a )= = sgn(a n )=s,, otherwse, (.9a) u (l),ma = ( ) l m (l )u (l),u (l ) u (l ),u (l ) u (l ), (.9b) u (l ) = u (l ) (l )u (l), u(l ) = u (l ) (l )u (l). Ths lmtng procedure s appled n the same adaptve way as BDF lmter.. A monotoncty-preservng lmter (MP lmter) []. Defne (.9c) medan(,y,z) = m(y,z ), (.) where m s the mnmod functon. We modfy the pont value u,mod = medan ( u,u mn,u ma ), (.) where u mn [ = ma u ma = mn mn(u (),u () [ ma(u (),u (),umd,umd ] ),mn(u (),u UL,u LC ) ] ),ma(u (),u UL,u LC ), (.a), (.b) and d = u () u() u (), d MX u MD = u LC = u () (.a) = m(d d,d d,d,d,d,d ), (.b) ( ) ( ) u () u () dmx, u UL = u () α u () u (), (.c) ( ) u () u () β dmx. (.d) The pont value u s modfed n a smlar (symmetrc) way. We take the parameters α= and β= n the numercal tests n the net secton, as suggested n []. where. A modfed MP lmter (MMP lmter) []. The moments are lmted to u (l),mod = φ u (l), l k, (.) φ = mn(, u () / mn u ), mn u = u () mnu h (,t). (.) I Ths lmter relaes the constrant of preservng monotoncty whle enforcng weaker constrants and s a sgn-preservng lmter.

7 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9. A shock-detecton technque by Krvodonova et al. (KXRCF shock-detector) []. Partton the boundary of a cell I nto two portons I and I, where the flow s nto ( v n<, n s the normal vector to I ) and out of ( v n> ) I, respectvely. The cell I s beleved to contan a dscontnuty f (u h I u h In )ds I k >. (.) I u h I Here I n s the neghbor of I on the sde of I and the norm s based on an element average n one dmenson. Ths KXRCF shock-detector functons as the frst part of the lmter, that s a troubledcell ndcator. It s one of the three recommended troubled-cell ndcators n the comparson of dfferent troubled-cell ndcators carred out by Qu and Shu []. It s also one of the best troubled-cell ndcators for the h- and r-adaptve RKDG methods presented n []. The good performance of ths shock-detector motvates us to use t to desgn new lmters. In ths paper, we add ths shock-detector to the orgnal lmters descrbed above to form new lmters and compare ther performance. Each new lmter works n the followng way. It frst uses KXRCF shock-detector to dentfy troubled-cells, and then apples the orgnal lmter only n these troubled-cells. We also combne ths shockdetector wth the WENO soluton reconstructon method ntroduced n [] and a smple WENO soluton reconstructon method recently ntroduced n [] (whch are both descrbed below), and the resultng lmters are denoted by WENO lmter and SWENO lmter, respectvely. For the sake of convenence, all the lmters n ths paper that nvolve KXRCF shock-detector are called by KXRCF-type lmters. If a KXRCF-type lmter s added to an orgnal lmter, t s denoted by addng (K) after the name of the orgnal lmter, for eample TVB-(K) lmter.. WENO soluton reconstructon method []. From (.) we can derve u (l) = l u h (,t)v () l ()d, l=,,k. (.) I If we use numercal quadratures to compute the ntegratons n (.), to reconstruct cell I s degrees of freedom u (l) (l=,,k), all we have to do s to reconstruct the pont values of u h at the quadrature ponts. As s ndcated n [], the followng partcular quadrature ponts are used: k=: two-pont Gauss quadrature ponts and ; k=: four-pont Gauss-Lobatto quadrature ponts,, and. Let us now reconstruct the pont value at some quadrature pont G. For j=,,k, we have a k-th degree polynomal reconstructon p j () such that j l p j ()d=u () j l, l=,,k. (.8) I j l

8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 We also have a k-th degree polynomal reconstructon Q() such that Q()d=u () l, l= k,,k. (.9) I l l Fnd the so called lnear weghts γ,,γ k, whch satsfy Q( G )= Then we compute the smoothness ndcators β j = k l= and the nonlnear weghts ω,,ω k, k j= γ j p j ( G ). (.) ( l l ) d, I l p j() j=,,k, (.) ω j = ω j j ω j, ω j = γ j, j=,,k. (.) (εβ j ) ε s a small number to avod the denomnator becomng zero. We use ε= n all computatons n ths paper. The fnal WENO appromaton s then gven by u WENO ( G )= k j= ω j p j ( G ). (.) 8. A smple WENO soluton reconstructon method []. Assume that we need to reconstruct the soluton on cell I. Denote the DG soluton polynomal of u on cells I, I, I as p (), p (), p (), respectvely. Denote p ()= p () p p, p ()= p () p p, (.) where p j = p j ()d, I We frst compute the smoothness ndcators β j = k l= j=,,. ( l l ) d, I l p j() j=,,. (.) For any lnear weghts γ, γ, γ, we then compute the nonlnear weghts ω, ω, ω ω j = ω j j ω j, Fnally, the fnal WENO reconstructon polynomal s gven by ω j = γ j, j=,,. (.) (εβ j ) p SWENO ()=ω p ()ω p ()ω p (). (.) We take ε=, γ =., γ =.998, γ =. n our numercal tests as n [].

9 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 Remark.. For the case of hyperbolc systems, all the lmtng procedures n ths secton are performed n local characterstc drectons. For detals we refer to [, 9]. The two-dmensonal case s not covered because of space lmtaton. We refer to the correspondng orgnal paper of each lmter. Numercal tests and comparsons In ths secton we provde a seres of numercal eamples to test and compare the lmters. The lmters are performed after each nner stage n the Runge-Kutta tme steppng. A postvty-preservng technque [] s used n all the test problems n order to avod negatve densty or negatve pressure durng the tme evoluton. We wll focus on the percentage of lmted cells and the control of spurous oscllatons n the comparson accordng to the functons of lmters. Also, we wll only plot the unform-mesh results obtaned wth a partcular choce of cell number N to save space, and only plot the dscontnuous regons (whch we are nterested n) n the soluton fgures for a better vew of detals. We use the one-dmensonal Euler equatons of gas dynamcs wth dfferent ntal data as our one-dmensonal test problems. The Euler equatons are ρ ρv E t ρv ρv p v(e p) =. (.) Here ρ s the densty, v s the velocty, E s the total energy, p s the pressure, related to the total energy by E= p/(γ )ρv / wth γ=.. Eample.. Sngle contact dscontnuty. The ntal condton s { (ηρ,,), f, (ρ,v,p)= (,,), otherwse. We choose η ρ = n (n=,,) n the numercal runnng but only the results for η ρ = are shown because the results for other choces of η ρ are smlar. The computatonal doman s [,]. In TVB- lmter, the parameter M=. We plot the densty solutons on [.,.] at T= n Fgs. and. In these fgures the sold lne s the reference eact soluton and each or represents average densty of a cell. In Fgs. and we can see that for the orgnal lmters, contact dscontnutes are well resolved wth TVB-, BDF, BSB and MP lmters whle oscllatons appear wth MMP lmter for both k = and k = cases. For the KXRCF-type lmters, there are oscllatons near the dscontnuty n all the cases of k=. For k= all the solutons are well resolved ecept that slght oscllatons are observed wth MMP(K) lmter. Compared to the orgnal lmters, addng KXRCF shock-detector causes slght oscllatons for TVB, BDF, BSB and MP lmters when k=.

10 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9.E.E TVB- TVB-(K) BDF BDF(K) E.E BSB BSB(K) MP MP(K) E.E MMP MMP(K) WENO SWENO Fgure : Sngle contact dscontnuty, densty solutons, η ρ =, N=, k=. In order to compare the lmtng effcency, we mark every lmted cell at each tme step and compute the average and mamum percentage of lmted cells through all tme steps. These data are tabulated n Table, n whch Ave and Ma denote the average and mamum percentage of lmted cells, respectvely, and those data derved wth

11 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9.E.E TVB- TVB-(K) BDF BDF(K) E.E BSB BSB(K) MP MP(K) E.E MMP MMP(K) WENO SWENO Fgure : Sngle contact dscontnuty, densty solutons, η ρ =, N=, k=. KXRCF-type lmters are marked wth (K). Ths table demonstrates the dfferent behavors of dfferent lmters for the same problem. In ths test problem BDF, BSB and MP lmters do much more lmtng than the other orgnal lmters. For KXRCF-type lmters, all of them do a low amount of lmtng, no matter how much lmtng ther cor-

12 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 Table : Sngle contact dscontnuty, average and mamum percentage of lmted cells, N =. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO SWENO respondng orgnal lmters do. For ths problem,usng KXRCF shock-detector to locate dscontnuous regons before lmtng reduces the soluton reconstructon work to a low amount. Eample.. Sngle shock. The ntal condton s (,,), f, (ρ,v,p)= ( ηp η p, η ) p.ηp.,η p, otherwse. We choose η p = n (n=,,) n the numercal runnng but only show the results for η p =. The computatonal doman s [,] and the parameter M= n TVB- lmter. We solve ths problem tll T =.. We plot the densty solutons on [, ] n Fgs. and. We observe slght oscllatons for BSB(K) lmter and obvous oscllatons for MMP and MMP(K) lmters n both k = and k = cases. The shocks are smeared more wth WENO lmter though t s oscllatory free, and SWENO lmter smears the most n ths test problem. Table gves average and mamum percentage of lmted cells. We agan see that BDF, BSB and MP lmters do much more lmtng whle all the other lmters do a comparable low amount of lmtng. Table : Sngle shock, average and mamum percentage of lmted cells, N =. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO SWENO....

13 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure : Sngle shock, densty solutons, η p =, N=, k=. Eample.. La problem []. The ntal condton s (ρ,v,p)= { (.,.98,.8), f, (.,,.), f >.

14 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure : Sngle shock, densty solutons, η p =, N=, k=. The computatonal doman s[,] and the parameter M= n TVB- lmter. We solve ths problem untl a smulaton tme of.. We plot the densty solutons on [.,.8] n Fgs. and. We can see from these fgures that addng KXRCF shock-detector brngs slght oscllatons to BDF, BSB and MP

15 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K).8.. BDF BDF(K) BSB BSB(K).. MP MP(K) MMP MMP(K).. WENO SWENO.. Fgure : La problem, densty solutons, N=, k=. lmters near the contact dscontnuty when k =. Oscllatons are also observed wth MMP and MMP(K) lmters when k= and wth MMP(K) and SWENO lmters when k=. We gve average and mamum percentage of lmted cells n Table. In ths test

16 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K).. BDF BDF(K) BSB BSB(K).. MP MP(K) MMP MMP(K).. WENO SWENO.. Fgure : La problem, densty solutons, N=, k=. problem BDF, BSB and MP lmters agan do much more lmtng than the other orgnal lmters. MMP lmter does the least lmtng but t does too lttle to control the oscllatons when k =. For KXRCF-type lmters, all of them do a consderable low amount of lmtng. We also see that addng KXRCF shock-detector to the orgnal lmters reduces the soluton reconstructon work.

17 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp Table : La problem, average and mamum percentage of lmted cells, N =. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO.... SWENO.8... Eample.. Shu-Osher problem [9]. Soluton of ths test problem contans both shocks and comple smooth regons. The ntal condton s (ρ,v,p)= { (.8,.99,.), f <, (.sn(),,), f. The computatonal doman s [,] and the parameter M = n TVB- lmter. We solve ths problem up to T=.8. Table gves data of average and mamum percentage of lmted cells. Ths table provdes the same nformaton as the prevous tables. We agan see that BDF, BSB and MP lmters do a lot of lmtng whle the others do much less. Results of densty solutons on[,.] are shown n Fgs. and 8. All the KXRCF-type lmters ntroduce oscllatons near the small jumps when k = whle all the orgnal lmters do not. We also notce that BDF(K) and BSB(K) lmters gve better appromatons n the comple smooth regon than BDF and BSB lmters, respectvely. Table : Shu-Osher problem, average and mamum percentage of lmted cells, N =. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO.8... SWENO...9.

18 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure : Shu-Osher problem, densty solutons, N=, k=. Eample.. The blast wave problem []. Ths classcal test problem nvolves nteracton of blast waves and ts ntal condton s gven by (,,), f <., (ρ,v,p)= (,,.), f. <.9, (,,), f.9.

19 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure 8: Shu-Osher problem, densty solutons, N=, k=. We compute the soluton tll T=.8. In the TVB- lmter, the parameter M=. Reflectve boundary condtons are appled to both ends. We gve the numercal results n Table and Fgs. 9 and (densty solutons on [.,.9] are shown). We can see agan that BDF, BSB and MP lmters result n too much lmtng. MMP and MMP(K) lmters do the least lmtng and gve the best resoluton of

20 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure 9: The blast wave problem, densty solutons, N=, k=. the shocks, but they ntroduce small oscllatons. Shocks are serously smeared and the solutons are poor wth WENO and SWENO lmters. Eample.. Two-dmensonal Burgers equaton. In ths last eample we consder the two-dmensonal Burgers equaton

21 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO Fgure : The blast wave problem, densty solutons, N=, k=. ( u ( u u t ) ) =,,y, (.) y wth the ntal condton u(,y,) =.sn(π(y)/) and perodc boundary condtons. The eact soluton s one-dmensonal dependng only on ξ = y; however, our

22 8 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp. -9 Table : The blast wave problem, average and mamum percentage of lmted cells, N=. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO.... SWENO Table : Two-dmensonal Burgers equaton, average and mamum percentage of lmted cells, 8 cells. k= k= Lmter Ave Ma Ave(K) Ma(K) Ave Ma Ave(K) Ma(K) TVB BDF BSB MP MMP WENO SWENO meshes are unformly rectangular n the (,y) coordnates, and thus ths eample s a truly two-dmensonal test problem. The parameter M= n the TVB- lmter. We compute the solutons wth 8 cells untl t=./π and plot the soluton on the dagonal cells n Fgs. and. We can see that all the lmters obtan satsfactory numercal appromatons wth sharp and nonoscllatory shock transtons ecept that overshoots are observed wth MMP, MMP(K) lmters for k = and MP, MP(K), MMP, MMP(K) lmters for k=. Table gves data of average and mamum percentage of lmted cells. We can see from ths table that all the orgnal lmters do a low amount of lmtng and the amount of lmtng done by KXRCF-type lmters s even lower. Concludng remarks In ths paper, we numercally study and compare multple lmters through a seres of classcal test problems. Detaled numercal results are presented n order to gan a better understandng of each lmter. The numercal results show that () TVB- lmter behaves consstently well n all eamples but t has a parameter to tune artfcally whch s a serous defect of ths lmter, that () BDF, BSB and MP lmters usually do too much lmtng, that () MMP lmter often causes oscllatons because t does too lttle lmtng,

23 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K).. WENO SWENO Fgure : Two-dmensonal Burgers equaton, soluton that cuts along the dagonal wth 8 cells, k =. that (v) WENO and SWENO lmters often ntroduce more smearng of jumps, that (v) usng KXRCF shock-detector to locate dscontnutes before lmtng can reduce the work of soluton reconstructon to a consderable low amount, but often brngs oscllatons when k=.

24 88 H. Zhu, Y. Cheng and J. Qu / Adv. Appl. Math. Mech., (), pp TVB- TVB-(K) BDF BDF(K) BSB BSB(K) MP MP(K) MMP MMP(K) WENO SWENO - - Fgure : Two-dmensonal Burgers equaton, soluton that cuts along the dagonal wth 8 cells, k =. Acknowledgments The research was partally supported by NSFC grant 9, 8,, NJUPT grant NY9 and ISTCP of Chna grant No. DFR. The authors would lke to thank the referees for the helpful suggestons.

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